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Franco Giannessi : “Every mathematician can do a true theorem : only a genius can make an important mistake”

Necessary conditions for non-intersection of collections of sets

  • Authors: Bui, Hoa , Kruger, Alexander
  • Date: 2022
  • Type: Text , Journal article
  • Relation: Optimization Vol. 71, no. 1 (2022), p. 165-196
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues studies of non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle. We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish new primal (slope) and dual (generalized separation) necessary conditions for these non-intersection properties. The results are applied to convergence analysis of alternating projections. © 2021 Informa UK Limited, trading as Taylor & Francis Group.

On (local) analysis of multifunctions via subspaces contained in graphs of generalized derivatives

  • Authors: Gfrerer, Helmut , Outrata, Jiri
  • Date: 2022
  • Type: Text , Journal article
  • Relation: Journal of Mathematical Analysis and Applications Vol. 508, no. 2 (2022), p.
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the graphically Lipschitzian mappings and thus a number of multifunctions, frequently arising in optimization and equilibrium problems. The developed theory makes use of new generalized derivatives, provides us with some calculus rules and reveals a number of interesting connections. In particular, it enables us to construct a modification of the semismooth* Newton method with improved convergence properties and to derive a generalization of Clarke's Inverse Function Theorem to multifunctions together with new efficient characterizations of strong metric (sub)regularity and tilt stability. © 2021 The Author(s)

Optimality conditions, approximate stationarity, and applications 'a story beyond lipschitzness

  • Authors: Kruger, Alexander , Mehlitz, Patrick
  • Date: 2022
  • Type: Text , Journal article
  • Relation: ESAIM - Control, Optimisation and Calculus of Variations Vol. 28, no. (2022), p.
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: Approximate necessary optimality conditions in terms of Frechet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland's variational principle, the fuzzy Frechet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established. © The authors.

A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure

Enlargements of the moreau–rockafellar subdifferential

Gateaux differentiability revisited

On the regularity of weak solutions of the boussinesq equations in besov spaces

Primal necessary characterizations of transversality properties

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Positivity Vol. 25, no. 2 (2021), p. 531-558
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of general nonlinear transversality properties of collections of sets and focuses on primal necessary (in some cases also sufficient) characterizations of the properties. We formulate geometric, metric and slope characterizations, particularly in the convex setting. The Hölder case is given a special attention. Quantitative relations between the nonlinear transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings as well as two nonlinear transversality properties of a convex set-valued mapping to a convex set in the range space are discussed. © 2020, Springer Nature Switzerland AG.

Transversality properties : primal sufficient conditions

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 29, no. 2 (2021), p. 221-256
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed. © 2020, Springer Nature B.V.

A uniform approach to hölder calmness of subdifferentials

  • Authors: Beer, Gerald , Cánovas, Maria , López, Marco , Parra, Juan
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Journal of Convex Analysis Vol. 27, no. 1 (2020), p.
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: For finite-valued convex functions f defined on the n-dimensional Euclidean space, we are interested in the set-valued mapping assigning to each pair (f, x) the subdifferential of f at x. Our approach is uniform with respect to f in the sense that it involves pairs of functions close enough to each other, but not necessarily around a nominal function. More precisely, we provide lower and upper estimates, in terms of Hausdorff excesses, of the subdifferential of one of such functions at a nominal point in terms of the subdifferential of nearby functions in a ball centered in such a point. In particular, we obtain the (1/2) - Hölder calmness of our mapping at a nominal pair (f, x) under the assumption that the subdifferential mapping viewed as a set-valued mapping from Rn to Rn with f fixed is calm at each point of {x} × ∂f(x). © Heldermann Verlag
  • Description: Funding details: Australian Research Council, ARC, DP160100854 Funding details: European Commission, EU Funding details: Ministerio de Economía y Competitividad, MINECO Funding details: Federación Española de Enfermedades Raras, FEDER Funding text 1: 

Nonlinear transversality of collections of sets : dual space necessary characterizations

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Journal of Convex Analysis Vol. 27, no. 1 (2020), p. 285-306
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of `good arrangements' of collections of sets in normed spaces near a point in their intersection. Our aim is to study general nonlinear transversality properties. We focus on dual space (subdifferential and normal cone) necessary characterizations of these properties. As an application, we provide dual necessary conditions for the nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe.
  • Description: The research was supported by the Australian Research Council, project DP160100854. The second author benefited from the support of the FMJH Program PGMO and from the support of EDF.

On computation of optimal strategies in oligopolistic markets respecting the cost of change

  • Authors: Outrata, Jiri , Valdman, Jan
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Mathematical Methods of Operations Research Vol. 92, no. 3 (2020), p. 489-509
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The paper deals with a class of parameterized equilibrium problems, where the objectives of the players do possess nonsmooth terms. The respective Nash equilibria can be characterized via a parameter-dependent variational inequality of the second kind, whose Lipschitzian stability, under appropriate conditions, is established. This theory is then applied to evolution of an oligopolistic market in which the firms adapt their production strategies to changing input costs, while each change of the production is associated with some “costs of change”. We examine both the Cournot-Nash equilibria as well as the two-level case, when one firm decides to take over the role of the Leader (Stackelberg equilibrium). The impact of costs of change is illustrated by academic examples. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

On the Aubin property of solution maps to parameterized variational systems with implicit constraints

  • Authors: Gfrerer, Helmut , Outrata, Jiri
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Optimization Vol. 69, no. 7-8 (2020), p. 1681-1701
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: In the paper, a new sufficient condition for the Aubin property to a class of parameterized variational systems is derived. In these systems, the constraints depend both on the parameter as well as on the decision variable itself and they include, e.g. parameter-dependent quasi-variational inequalities and implicit complementarity problems. The result is based on a general condition ensuring the Aubin property of implicitly defined multifunctions which employs the recently introduced notion of the directional limiting coderivative. Our final condition can be verified, however, without an explicit computation of these coderivatives. The procedure is illustrated by an example. © 2019, © 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
  • Description: The research of the first author was supported by the Austrian Science Fund (FWF) under grant P29190-N32. The research of the second author was supported by the Grant Agency of the Czech Republic, Project 17-04301S and the Australian Research Council, Project 10.13039/501100000923DP160100854.

Set-valued orthogonality and nearness

Convexity and closedness in stable robust duality

  • Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel
  • Date: 2019
  • Type: Text , Journal article
  • Relation: Optimization Letters Vol. 13, no. 2 (2019), p. 325-339
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The paper deals with optimization problems with uncertain constraints and linear perturbations of the objective function, which are associated with given families of perturbation functions whose dual variable depends on the uncertainty parameters. More in detail, the paper provides characterizations of stable strong robust duality and stable robust duality under convexity and closedness assumptions. The paper also reviews the classical Fenchel duality of the sum of two functions by considering a suitable family of perturbation functions.

Metric regularity relative to a cone

On semiregularity of mappings

  • Authors: Cibulka, Radek , Fabian, Marian , Kruger, Alexander
  • Date: 2019
  • Type: Text , Journal article
  • Relation: Journal of Mathematical Analysis and Applications Vol. 473, no. 2 (2019), p. 811-836
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature. We provide a self-contained material gathering and extending the existing theory on the topic. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point is shown. In particular cases, we derive necessary and/or sufficient conditions of both primal and dual type. We illustrate the importance of semiregularity in the convergence analysis of an inexact Newton-type scheme for generalized equations with not necessarily differentiable single-valued part. © 2019 Elsevier Inc.

About extensions of the extremal principle

  • Authors: Bui, Hoa , Kruger, Alexander
  • Date: 2018
  • Type: Text , Journal article
  • Relation: Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 215-242
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: In this paper, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets, and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions, and expand the applicability of the generalized separation results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity, and Lipschitz-like properties of set-valued mappings. © 2018, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

About intrinsic transversality of pairs of sets

  • Authors: Kruger, Alexander
  • Date: 2018
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 26, no. 1 (2018), p. 111-142
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The article continues the study of the ‘regular’ arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimization as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterizations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case.